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Theorem prnz 4777
Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1 𝐴 ∈ V
Assertion
Ref Expression
prnz {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3 𝐴 ∈ V
21prid1 4762 . 2 𝐴 ∈ {𝐴, 𝐵}
32ne0ii 4344 1 {𝐴, 𝐵} ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  wne 2940  Vcvv 3480  c0 4333  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629
This theorem is referenced by:  opnz  5478  propssopi  5513  fiint  9366  fiintOLD  9367  wilthlem2  27112  upgrbi  29110  wlkvtxiedg  29643  shincli  31381  chincli  31479  constrextdg2lem  33789  spr0nelg  47463  sprvalpwn0  47470
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